Monte Carlo simulations of (very) simple quantum systems

• hydrogen molecule
• triplet state of helium atom (orthohelium)

The results listed here were obtained with the so-called diffusion Monte Carlo (DMC) method that is based on two observations:

• when viewed from the right angle, the Schrödinger equation looks a lot like a many-dimensional diffusion equation
• the diffusion can be modeled on a computer as a Brownian motion of particles through the many-dimensional space

It is remarkable that an entirely classical stochastic process – the Brownian motion – can be so helpful for solving equations of quantum mechanics. And not only in the elementary two-electron systems shown here, but also in a variety of pretty serious and highly non-trivial settings. Examples of the serious applications as well as technical details of the method can be found, for instance, in the following papers:

• J. Kolorenc and L. Mitas, Applications of quantum Monte Carlo methods in condensed systems, Rep. Prog. Phys. 74, 026502 (2011) [arXiv:1010.4992]
• W. M. C. Foulkes, L. Mitas, R. J. Needs, and G. Rajagopal, Quantum Monte Carlo simulations of solids, Rev. Mod. Phys. 73, 33–83 (2001) [a copy on LM's website]
• P. J. Reynolds, D. M. Ceperley, B. J. Alder, and W. A. Lester, Jr., Fixed-node quantum Monte Carlo for molecules, J. Chem. Phys. 77, 5593–5603 (1982)
• I. Kosztin, B. Faber, and K. Schulten, Introduction to the diffusion Monte Carlo method, Am. J. Phys. 64, 633–644 (1996)

H₂ molecule

The DMC simulation is started from an approximate solution, in this particular case it was the Heitler–London ansatz

where s_A and s_B are the lowest s-orbitals at the first and the second proton [W. Heitler and F. London, Z. Physik 44, 455 (1927); Y. Sugiura, Z. Physik 45, 484 (1927)]. After some time, the auxiliary diffusion reaches a stationary state that corresponds to the exact ground state of the molecule. The blue line in the picture shows the starting point (the Heitler–London approximation), the red points indicate the DMC results.

The ground-state energy is only one of many physical quantities that can be extracted from a DMC simulation. For example, the electron density can be calculated to visualize the covalent chemical bond. The following figure shows the density at the equilibrium proton distance 1.4 bohr.

Orthohelium

Calculation of the parahelium ground state (antiparallel electron spins, i.e. spin singlet) almost does not differ from the hydrogen molecule above. The orthohelium (parallel electron spins, i.e. spin triplet) is more interesting because the spatial part of the electron wave function is antisymmetric (changes sign when the electrons are interchanged) that makes the correspondence between quantum mechanics and classical diffusion rather problematic. However, it can be done if we know position of nodes.

Details of two calculations with different time steps:

• ground-state energy: −2.1753 ± 2·10^-4 Hartree
• importance sampling with the wave function
  ψ(r₁,r₂) = s₁(r₁)s₂(r₂) − s₁(r₂)s₂(r₁),
  where s₁ and s₂ are hydrogen-like s-orbitals from the first and the second shell, respectively. It is the Pauli exclusion principle that forbids the electrons to sit both in s₁ when having parallel spins.
• number of walkers: 10⁶,
• time step: 0.0025,
• projection/relaxation time: 30000 steps,
• time of energy collection: following 4000 steps.

• ground-state energy: −2.1755 ± 2·10^-4 Hartree
• time step: 0.005,
• projection/relaxation time: 15000 steps.

Demanding such a high precision has a reason in distinguishing the DMC simulation from quite simple variational calculation that takes the above wave function and optimizes the core charges seen by the electrons. This procedure already gives very nice result −2.1666 Hartree with charges Z₁=1.99363 for the inner electron (in its case the core is not screened at all) and Z₂=1.55093 for the outer electron, see my Mathematica notebook if interested (do you know MathReader ?).